68

последняя 8

предпоследняя 6

Вариант 3


SO,SN,OWNWS \lor O, S \rarr N, O \rarr W \rightarrowtail N \lor W

4

M={1,2,3,...,20}M = \{ 1, 2, 3, ..., 20 \}

a

C(x)A(x)C(x) \rarr A(x)

C(x)A(x)\overline {C(x)} \lor A(x)

"x - не простое число, либо оно не делится на 5"

C(x)A(x)\overline {C(x) \land \overline {A(x)}}

"x - не является простым числом, делящимся на 5"

в множестве M только 5 - одновременно и простое число и делящееся на 5

тогда множество истинности:

R=M{5}R = M \setminus \{5\}

b

D(x)C(x)D(x) \rarr \overline C(x)

D(x)C(x)\overline D(x) \lor \overline C(x)

D(x)C(x)\overline {D(x) \land C(x)}

"x не является простым числом, кратным 3"

простое число, делящееся на 3 - это 3

тогда множество истинности:

R=M{3}R = M \setminus \{3\}

с

(A(x)C(x))D(x)(A(x) \land C(x)) \rarr \overline D(x)

"если x - простое число, не делящееся на 5, то оно не кратно 3"

A(x)C(x)D(x)\overline {A(x) \land C(x)} \lor \overline D(x)

A(x)C(x)D(x)\overline {A(x) \land C(x) \land D(x)}

"x не является простым числом, которое делится на 3, но не на 5"

простые числа, которые не делятся на 5:

если оно еще и делится на 3, то это точно 3.

тогда множество истинности:

R=M{3}R = M \setminus \{3\}

5

a

x(A(x)B(y))&y(A(x)(B(y)C(z)))z(A(x)C(z))\forall x (A(x) \rarr B(y)) \& \forall y (A(x) \rarr (B(y) \rarr C(z))) \rarr \exists z (A(x) \rarr C(z))

x(A(x)B(y))&y(A(x)(B(y)C(z)))z(A(x)C(z))\forall x (\overline A(x) \lor B(y)) \& \forall y (A(x) \rarr (\overline B(y) \lor C(z))) \rarr \exists z (\overline A(x) \lor C(z))

x(A(x)B(y))&y(A(x)B(y)C(z))z(A(x)C(z))\forall x (\overline A(x) \lor B(y)) \& \forall y (\overline A(x) \lor \overline B(y) \lor C(z)) \rarr \exists z (\overline A(x) \lor C(z))

x(A(x)B(y))&y(A(x)B(y)C(z))z(A(x)C(z))\overline {\forall x (\overline A(x) \lor B(y)) \&\forall y (\overline A(x) \lor \overline B(y) \lor C(z))} \lor \exists z (\overline A(x) \lor C(z))

x(A(x)B(y))y(A(x)B(y)C(z))z(A(x)C(z))\overline {\forall x (\overline A(x) \lor B(y))} \lor \overline{\forall y (\overline A(x) \lor \overline B(y) \lor C(z))} \lor \exists z (\overline A(x) \lor C(z))

x(A(x)B(y))y(A(x)B(y)C(z))z(A(x)C(z))\exist x \overline {(\overline A(x) \lor B(y))} \lor \exist y \overline{(\overline A(x) \lor \overline B(y) \lor C(z))} \lor \exists z (\overline A(x) \lor C(z))

x(A(x)B(y))y(A(x)B(y)C(z))z(A(x)C(z))\exist x (A(x) \land \overline B(y)) \lor \exist y (\overline{\overline A(x) \lor \overline B(y)} \land \overline C(z)) \lor \exists z (\overline A(x) \lor C(z))

x(A(x)B(y))y(A(x)B(y)C(z))z(A(x)C(z))\exist x (A(x) \land \overline B(y)) \lor \exist y (A(x) \land B(y) \land \overline C(z)) \lor \exists z (\overline A(x) \lor C(z))

xA(x)B(y)yB(y)A(x)C(z)A(x)zC(z)\exist x A(x) \land \overline B(y) \lor \exist y B(y) \land A(x) \land \overline C(z) \lor \overline A(x) \lor \exists z C(z)

b

x(¬A(x)x(¬C(x)))x((C(x)A(x))\forall x(\lnot A(x) \rarr \exists x (\lnot C(x) )) \rarr \forall x ((C(x) \rarr A(x))

x(¬A(x)x(¬C(x)))x(C(x)A(x))\forall x(\overline {\lnot A(x)} \lor \exists x (\lnot C(x) )) \rarr \forall x (\overline C(x) \lor A(x))

x(A(x)x(¬C(x)))x(C(x)A(x))\overline {\forall x (A(x) \lor \exists x (\lnot C(x) ))} \lor \forall x (\overline C(x) \lor A(x))

xA(x)x(¬C(x))x(C(x)A(x))\overline {\forall x A(x)} \land \overline{\exists x (\lnot C(x) )} \lor \forall x (\overline C(x) \lor A(x))

xA(x)x(¬C(x))x(C(x)A(x))\exists x \overline {A(x)} \land \forall x \overline{(\lnot C(x) )} \lor \forall x (\overline C(x) \lor A(x))

xA(x)xC(x)x(C(x)A(x))\exists x \overline {A(x)} \land \forall x C(x) \lor \forall x (\overline C(x) \lor A(x))

c

(x(¬A(x))x(¬B(x)))(¬B(x)A(x))(\exists x (\lnot A(x)) \rarr \forall x (\lnot B(x))) \rarr (\lnot B(x) \lor A(x))

(xA(x)xB(x))(¬B(x)A(x))(\overline{\forall x A(x)} \rarr \overline{\exist x B(x)}) \rarr (\lnot B(x) \lor A(x))

(xA(x)xB(x))(¬B(x)A(x))(\forall x A(x) \lor \overline{\exist x B(x)}) \rarr (\lnot B(x) \lor A(x))

xA(x)xB(x)(¬B(x)A(x))\overline {\forall x A(x) \lor \overline{\exist x B(x)}} \lor (\lnot B(x) \lor A(x))

xA(x)xB(x)B(x)A(x)\overline {\forall x A(x)} \land {\exist x B(x)} \lor \overline B(x) \lor A(x)

xA(x)xB(x)B(x)A(x)\exists x \overline {A(x)} \land {\exist x B(x)} \lor \overline B(x) \lor A(x)

6

a

(x(¬A(x))x(¬B(x)))(¬B(x)A(x))(\exists x (\lnot A(x)) \rarr \forall x (\lnot B(x))) \rarr (\lnot B(x) \lor A(x))

(x(¬A(x))x(¬B(x)))(¬B(x)A(x))(\overline{\exists x (\lnot A(x))} \lor \forall x (\lnot B(x))) \rarr (\lnot B(x) \lor A(x))

(xA(x)x(¬B(x)))(¬B(x)A(x))(\forall x A(x) \lor \forall x (\lnot B(x))) \rarr (\lnot B(x) \lor A(x))

(xA(x)x(¬B(x)))(¬B(x)A(x))\overline{(\forall x A(x) \lor \forall x (\lnot B(x)))} \lor (\lnot B(x) \lor A(x))

xA(x)x(¬B(x))(¬B(x)A(x))\overline{\forall x A(x)} \land \overline{\forall x (\lnot B(x))} \lor (\lnot B(x) \lor A(x))

xA(x)xB(x)B(x)A(x)\exist x \overline{A(x)} \land \exist x B(x) \lor \overline B(x) \lor A(x)

yA(y)zB(z)B(x)A(x)\exist y \overline{A(y)} \land \exist z B(z) \lor \overline B(x) \lor A(x)

y(A(y)zB(z))B(x)A(x)\exist y (\overline{A(y)} \land \exist z B(z)) \lor \overline B(x) \lor A(x)

y(zB(z)A(y))B(x)A(x)\exist y (\exist z B(z) \land \overline{A(y)}) \lor \overline B(x) \lor A(x)

y(z(B(z)A(y)))B(x)A(x)\exist y (\exist z (B(z) \land \overline{A(y)})) \lor \overline B(x) \lor A(x)

yz(B(z)A(y)B(x)A(x))\exist y \exist z (B(z) \land \overline{A(y)} \lor \overline B(x) \lor A(x))

b

(x(A(x)y(B(y)C(z))))&(¬y(D(x;y)F(z)))(\forall x (A(x) \rarr \forall y (B(y) \rarr C(z)))) \, \& \, (\lnot \forall y (D(x;y) \rarr F(z)))

(x(A(x)y(B(y)C(z))))&(¬y(D(x;y)F(z)))(\forall x (A(x) \rarr \forall y (\overline B(y) \lor C(z)))) \, \& \, (\lnot \forall y (D(x;y) \rarr F(z)))

(x(A(x)(yB(y)C(z))))&(¬y(D(x;y)F(z)))(\forall x (A(x) \rarr (\forall y \overline B(y) \lor C(z)))) \, \& \, (\lnot \forall y (D(x;y) \rarr F(z)))

(x(A(x)(yB(y)C(z))))&(y¬(D(x;y)F(z)))(\forall x (\overline A(x) \lor (\forall y \overline B(y) \lor C(z)))) \, \& \, (\exist y \lnot (D(x;y) \rarr F(z)))

(xA(x)(yB(y)C(z)))&(y¬(D(x;y)F(z)))(\forall x \overline A(x) \lor (\forall y \overline B(y) \lor C(z))) \, \& \, (\exist y \lnot (\overline D(x;y) \lor F(z)))

(xA(x)(yB(y)C(z)))&(y(D(x;y)F(z)))(\forall x \overline A(x) \lor (\forall y \overline B(y) \lor C(z))) \, \& \, (\exist y (D(x;y) \land \overline F(z)))

(xA(x)yB(y)C(z))yD(x;y)F(z)(\forall x \overline A(x) \lor \forall y \overline B(y) \lor C(z)) \land \exist y D(x;y) \land \overline F(z)

(uA(u)tB(t)C(z))yD(x;y)F(z)(\forall u \overline A(u) \lor \forall t \overline B(t) \lor C(z)) \land \exist y D(x;y) \land \overline F(z)

y(D(x;y)F(z)(uA(u)tB(t)C(z)))\exists y (D(x;y) \land \overline F(z) \land (\forall u \overline A(u) \lor \forall t \overline B(t) \lor C(z)))

y(D(x;y)F(z)(u(A(u)tB(t)C(z))))\exists y (D(x;y) \land \overline F(z) \land (\forall u (\overline A(u) \lor \forall t \overline B(t) \lor C(z))))

y(D(x;y)F(z)(u(tB(t)A(u)C(z))))\exists y (D(x;y) \land \overline F(z) \land (\forall u (\forall t \overline B(t) \lor \overline A(u) \lor C(z))))

y(ut(D(x;y)F(z)(B(t)A(u)C(z))))\exists y (\forall u \forall t (D(x;y) \land \overline F(z) \land (\overline B(t) \lor \overline A(u) \lor C(z))))

yut(D(x;y)F(z)(B(t)A(u)C(z)))\exists y \forall u \forall t (D(x;y) \land \overline F(z) \land (\overline B(t) \lor \overline A(u) \lor C(z)))

c

(x(B(x)y(A(y))))y(A(x)C(y))¬C(y)B(x)(\forall x (B(x) \rarr \exists y (A(y)))) \land \exists y (A(x) \rarr C(y)) \rarr \lnot C(y) \land B(x)

(x(B(x)y(A(y))))y(A(x)C(y))¬C(y)B(x)\overline {(\forall x (\overline B(x) \lor \exists y (A(y)))) \land \exists y (\overline A(x) \lor C(y))} \lor \lnot C(y) \land B(x)

(xB(x)yA(y))(yC(y)A(x))C(y)B(x)\overline {(\forall x \overline B(x) \lor \exists y A(y)) \land (\exists y C(y) \lor \overline A(x)) } \lor \overline C(y) \land B(x)

(xB(x)yA(y))(yC(y)A(x))C(y)B(x)\overline {(\forall x \overline B(x) \lor \exists y A(y))} \lor \overline{ (\exists y C(y) \lor \overline A(x)) } \lor \overline C(y) \land B(x)

xB(x)yA(y)yC(y)A(x)C(y)B(x)\overline {\forall x \overline B(x)} \land \overline { \exists y A(y)} \lor \overline{ \exists y C(y)} \land A(x) \lor \overline C(y) \land B(x)

xB(x)yA(y)yC(y)A(x)C(y)B(x)\exists x B(x) \land \forall y \overline {A(y)} \lor \forall y \overline{C(y)} \land A(x) \lor \overline C(y) \land B(x)

xy(B(x)A(y))y(C(y)A(x))(C(y)B(x))\exists x \forall y (B(x) \land \overline {A(y)}) \lor \forall y (\overline{C(y)} \land A(x)) \lor (\overline C(y) \land B(x))

xy(B(x)A(y))t(C(t)A(x))(C(y)B(x))\exists x \forall y (B(x) \land \overline {A(y)}) \lor \forall t (\overline{C(t)} \land A(x)) \lor (\overline C(y) \land B(x))

xy(B(x)A(y))t(C(t)A(x)(C(y)B(x)))\exists x \forall y (B(x) \land \overline {A(y)}) \lor \forall t (\overline{C(t)} \land A(x) \lor (\overline C(y) \land B(x)))

uvt(B(u)A(v)(C(t)A(x)(C(y)B(x))))\exists u \forall v \forall t (B(u) \land \overline {A(v)} \lor (\overline{C(t)} \land A(x) \lor (\overline C(y) \land B(x))))

i'm really sorry...